Let n, k, t be positive integers. What is the maximum number of arcs in a digraph on n vertices in which there are at most t distinct walks of length k with the same endpoints? Determine the extremal digraphs attaining the maximum number. When \(t=1\), the problem has been studied by Wu, by Huang and Zhan, by Huang, Lyu and Qiao, by Lyu in four papers, and they solved all the cases but \(k=3\). For \(t\ge 2\), Huang and Lyu proved that the maximum number is equal to \(n(n-1)/2\) and the extremal digraph is the transitive tournament when \(n\ge 6t+2\) and \(k\ge n-1\). They also discussed the maximum number for the case \(n=k+2,k+3,k+4\). In this paper, we solve the problem for the case \(k\ge 6t+1\) and \(n\ge k+5\), and we also characterize the structures of the extremal digraphs for \(n=k+2,k+3,k+4\).

令n，  k，  t为正整数。在n个顶点上的图上最多有多少条弧，其中最多有t个不同的步长，且长度k相同，端点相同？确定达到最大数量的极值图。当\（t = 1 \）时，吴，黄和詹，黄，吕，乔，吕在四篇论文中研究了问题，他们解决了除（（k = 3 \）之外的所有情况。对于\（t \ ge 2 \），Huang和Lyu证明最大数等于\（n（n-1）/ 2 \）并且当\（n \ ge 6t + 2 \）和\（k \ ge n-1 \）。他们还讨论了情况\（n = k + 2，k + 3，k + 4 \）的最大数目。在本文中，我们解决了\（k \ ge 6t + 1 \）和\（n \ ge k + 5 \）的问题，并刻画了\（n = k + 2，k + 3，k + 4 \）。